In the usual approach, a Chern-Simons density is defined as the \(D^{\rm th}\)  component of
\( \bar{\Omega }^M\) ,
which results in a YM theory in a \(d=2n-1\)  odd-dimensional spacetimeAt no point is a metric involved in this construction; this is a topological theory..
However,
the approach in [1]}, [2]}
(see also the
Refs. [3]}, [4]}
and the Appendix A of Ref. [5]})
introduces an extra-step, by considering first the dimensional descent of the CP density (REF )
to some intermediate dimension \(d<D=2n\) ,
which can be odd or even.
As usual with gauge fields,
the relics of the gauge connection on the co-dimension(s) are Higgs scalar(s).
The remarkable property of the resulting density (dubbed now
Higgs-CP density \(\Omega _{\rm HCP}\) , being given in terms of both the residual gauge field and the Higgs scalar \(\Phi \) ),
is that, like the original CP density, it
is also a \(total\ divergence\) ,
\(\Omega _{\rm HCP}=\nabla _i W^i\ ,\quad i=1,.., d.\) 
